Many have argued that various features of moral disagreements create problems for cognitivism about moral judgment, but these arguments have been shown to fail. In this paper, I articulate a new problem for cognitivism that derives from features of our responses to moral disagreement. I argue that cognitivism entails that one of the following two claims is false: (1) a mental state is a belief only if it tracks changes in perceived evidence; (2) it is intelligible to make moral judgments that do not track changes in perceived evidence. I explain that there is a good case that (1) holds such that we should prefer theories that do not entail the negation of (1). And I argue that the seeming intelligibility of entirely intransigent responses to peer disagreement about moral issues shows us that there is a good case that (2) holds.